\(\int \frac {(f+g x)^2}{(a+b \log (c (d+e x)^n))^3} \, dx\) [100]

Optimal result

Integrand size = 24, antiderivative size = 351 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\frac {e^{-\frac {a}{b n}} (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac {4 e^{-\frac {2 a}{b n}} g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^3 n^3}+\frac {9 e^{-\frac {3 a}{b n}} g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{2 b^3 e^3 n^3}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \]

[Out]

1/2*(-d*g+e*f)^2*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b^3/e^3/exp(a/b/n)/n^3/((c*(e*x+d)^n)^(1/n))+4*g*(-d*g+
e*f)*(e*x+d)^2*Ei(2*(a+b*ln(c*(e*x+d)^n))/b/n)/b^3/e^3/exp(2*a/b/n)/n^3/((c*(e*x+d)^n)^(2/n))+9/2*g^2*(e*x+d)^
3*Ei(3*(a+b*ln(c*(e*x+d)^n))/b/n)/b^3/e^3/exp(3*a/b/n)/n^3/((c*(e*x+d)^n)^(3/n))-1/2*(e*x+d)*(g*x+f)^2/b/e/n/(
a+b*ln(c*(e*x+d)^n))^2+(-d*g+e*f)*(e*x+d)*(g*x+f)/b^2/e^2/n^2/(a+b*ln(c*(e*x+d)^n))-3/2*(e*x+d)*(g*x+f)^2/b^2/
e/n^2/(a+b*ln(c*(e*x+d)^n))

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2447, 2446, 2436, 2337, 2209, 2437, 2347} \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\frac {4 g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^3 n^3}+\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac {9 g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac {(d+e x) (f+g x) (e f-d g)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

[In]

Int[(f + g*x)^2/(a + b*Log[c*(d + e*x)^n])^3,x]

[Out]

((e*f - d*g)^2*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(2*b^3*e^3*E^(a/(b*n))*n^3*(c*(d + e
*x)^n)^n^(-1)) + (4*g*(e*f - d*g)*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b^3*e^3*E^
((2*a)/(b*n))*n^3*(c*(d + e*x)^n)^(2/n)) + (9*g^2*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*
n)])/(2*b^3*e^3*E^((3*a)/(b*n))*n^3*(c*(d + e*x)^n)^(3/n)) - ((d + e*x)*(f + g*x)^2)/(2*b*e*n*(a + b*Log[c*(d
+ e*x)^n])^2) + ((e*f - d*g)*(d + e*x)*(f + g*x))/(b^2*e^2*n^2*(a + b*Log[c*(d + e*x)^n])) - (3*(d + e*x)*(f +
g*x)^2)/(2*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n]))

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2337

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2446

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)), x_Symbol] :> Int[ExpandIn
tegrand[(f + g*x)^q/(a + b*Log[c*(d + e*x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && IGtQ[q, 0]

Rule 2447

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(d
+ e*x)*(f + g*x)^q*((a + b*Log[c*(d + e*x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Dist[(q + 1)/(b*n*(p + 1)), I
nt[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Dist[q*((e*f - d*g)/(b*e*n*(p + 1))), Int[(f + g*x
)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && LtQ[p, -1] && GtQ[q, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {3 \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{2 b n}-\frac {(e f-d g) \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{b e n} \\ & = -\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {9 \int \frac {(f+g x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{2 b^2 n^2}-\frac {(2 (e f-d g)) \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e n^2}-\frac {(3 (e f-d g)) \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e n^2}+\frac {(e f-d g)^2 \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2} \\ & = -\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {9 \int \left (\frac {(e f-d g)^2}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {2 g (e f-d g) (d+e x)}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g^2 (d+e x)^2}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx}{2 b^2 n^2}-\frac {(2 (e f-d g)) \int \left (\frac {e f-d g}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g (d+e x)}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx}{b^2 e n^2}-\frac {(3 (e f-d g)) \int \left (\frac {e f-d g}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g (d+e x)}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx}{b^2 e n^2}+\frac {(e f-d g)^2 \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2} \\ & = -\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left (9 g^2\right ) \int \frac {(d+e x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{2 b^2 e^2 n^2}-\frac {(2 g (e f-d g)) \int \frac {d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}-\frac {(3 g (e f-d g)) \int \frac {d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}+\frac {(9 g (e f-d g)) \int \frac {d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}-\frac {\left (2 (e f-d g)^2\right ) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}-\frac {\left (3 (e f-d g)^2\right ) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}+\frac {\left (9 (e f-d g)^2\right ) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{2 b^2 e^2 n^2}+\frac {\left ((e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3} \\ & = \frac {e^{-\frac {a}{b n}} (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^3 e^3 n^3}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left (9 g^2\right ) \text {Subst}\left (\int \frac {x^2}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 b^2 e^3 n^2}-\frac {(2 g (e f-d g)) \text {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac {(3 g (e f-d g)) \text {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac {(9 g (e f-d g)) \text {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac {\left (2 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac {\left (3 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac {\left (9 (e f-d g)^2\right ) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 b^2 e^3 n^2} \\ & = \frac {e^{-\frac {a}{b n}} (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^3 e^3 n^3}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left (9 g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 b^2 e^3 n^3}-\frac {\left (2 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac {\left (3 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}+\frac {\left (9 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac {\left (2 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac {\left (3 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}+\frac {\left (9 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 b^2 e^3 n^3} \\ & = \frac {e^{-\frac {a}{b n}} (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac {4 e^{-\frac {2 a}{b n}} g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^3 n^3}+\frac {9 e^{-\frac {3 a}{b n}} g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{2 b^3 e^3 n^3}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\frac {e^{-\frac {3 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (e^{\frac {2 a}{b n}} (e f-d g)^2 \left (c (d+e x)^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-8 e^{\frac {a}{b n}} g (-e f+d g) (d+e x) \left (c (d+e x)^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+9 g^2 (d+e x)^2 \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-b e e^{\frac {3 a}{b n}} n \left (c (d+e x)^n\right )^{3/n} (f+g x) \left (b e n (f+g x)+a (e f+2 d g+3 e g x)+b (2 d g+e (f+3 g x)) \log \left (c (d+e x)^n\right )\right )\right )}{2 b^3 e^3 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

[In]

Integrate[(f + g*x)^2/(a + b*Log[c*(d + e*x)^n])^3,x]

[Out]

((d + e*x)*(E^((2*a)/(b*n))*(e*f - d*g)^2*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)
]*(a + b*Log[c*(d + e*x)^n])^2 - 8*E^(a/(b*n))*g*(-(e*f) + d*g)*(d + e*x)*(c*(d + e*x)^n)^n^(-1)*ExpIntegralEi
[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log[c*(d + e*x)^n])^2 + 9*g^2*(d + e*x)^2*ExpIntegralEi[(3*(a +
b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log[c*(d + e*x)^n])^2 - b*e*E^((3*a)/(b*n))*n*(c*(d + e*x)^n)^(3/n)*(f +
g*x)*(b*e*n*(f + g*x) + a*(e*f + 2*d*g + 3*e*g*x) + b*(2*d*g + e*(f + 3*g*x))*Log[c*(d + e*x)^n])))/(2*b^3*e^3
*E^((3*a)/(b*n))*n^3*(c*(d + e*x)^n)^(3/n)*(a + b*Log[c*(d + e*x)^n])^2)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.73 (sec) , antiderivative size = 6545, normalized size of antiderivative = 18.65

method	result	size
risch	\(\text {Expression too large to display}\)	\(6545\)

[In]

int((g*x+f)^2/(a+b*ln(c*(e*x+d)^n))^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1090 vs. \(2 (344) = 688\).

Time = 0.31 (sec) , antiderivative size = 1090, normalized size of antiderivative = 3.11 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((g*x+f)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="fricas")

[Out]

1/2*(8*(a^2*e*f*g - a^2*d*g^2 + (b^2*e*f*g - b^2*d*g^2)*n^2*log(e*x + d)^2 + (b^2*e*f*g - b^2*d*g^2)*log(c)^2
+ 2*((b^2*e*f*g - b^2*d*g^2)*n*log(c) + (a*b*e*f*g - a*b*d*g^2)*n)*log(e*x + d) + 2*(a*b*e*f*g - a*b*d*g^2)*lo
g(c))*e^((b*log(c) + a)/(b*n))*log_integral((e^2*x^2 + 2*d*e*x + d^2)*e^(2*(b*log(c) + a)/(b*n))) + (a^2*e^2*f
^2 - 2*a^2*d*e*f*g + a^2*d^2*g^2 + (b^2*e^2*f^2 - 2*b^2*d*e*f*g + b^2*d^2*g^2)*n^2*log(e*x + d)^2 + (b^2*e^2*f
^2 - 2*b^2*d*e*f*g + b^2*d^2*g^2)*log(c)^2 + 2*((b^2*e^2*f^2 - 2*b^2*d*e*f*g + b^2*d^2*g^2)*n*log(c) + (a*b*e^
2*f^2 - 2*a*b*d*e*f*g + a*b*d^2*g^2)*n)*log(e*x + d) + 2*(a*b*e^2*f^2 - 2*a*b*d*e*f*g + a*b*d^2*g^2)*log(c))*e
^(2*(b*log(c) + a)/(b*n))*log_integral((e*x + d)*e^((b*log(c) + a)/(b*n))) - (b^2*d*e^2*f^2*n^2 + (b^2*e^3*g^2
*n^2 + 3*a*b*e^3*g^2*n)*x^3 + ((2*b^2*e^3*f*g + b^2*d*e^2*g^2)*n^2 + (4*a*b*e^3*f*g + 5*a*b*d*e^2*g^2)*n)*x^2
+ (a*b*d*e^2*f^2 + 2*a*b*d^2*e*f*g)*n + ((b^2*e^3*f^2 + 2*b^2*d*e^2*f*g)*n^2 + (a*b*e^3*f^2 + 6*a*b*d*e^2*f*g
+ 2*a*b*d^2*e*g^2)*n)*x + (3*b^2*e^3*g^2*n^2*x^3 + (4*b^2*e^3*f*g + 5*b^2*d*e^2*g^2)*n^2*x^2 + (b^2*e^3*f^2 +
6*b^2*d*e^2*f*g + 2*b^2*d^2*e*g^2)*n^2*x + (b^2*d*e^2*f^2 + 2*b^2*d^2*e*f*g)*n^2)*log(e*x + d) + (3*b^2*e^3*g^
2*n*x^3 + (4*b^2*e^3*f*g + 5*b^2*d*e^2*g^2)*n*x^2 + (b^2*e^3*f^2 + 6*b^2*d*e^2*f*g + 2*b^2*d^2*e*g^2)*n*x + (b
^2*d*e^2*f^2 + 2*b^2*d^2*e*f*g)*n)*log(c))*e^(3*(b*log(c) + a)/(b*n)) + 9*(b^2*g^2*n^2*log(e*x + d)^2 + b^2*g^
2*log(c)^2 + 2*a*b*g^2*log(c) + a^2*g^2 + 2*(b^2*g^2*n*log(c) + a*b*g^2*n)*log(e*x + d))*log_integral((e^3*x^3
+ 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*e^(3*(b*log(c) + a)/(b*n))))*e^(-3*(b*log(c) + a)/(b*n))/(b^5*e^3*n^5*log(e*
x + d)^2 + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3 + 2*(b^5*e^3*n^4*log(c) + a*b^4*e^3
*n^4)*log(e*x + d))

Sympy [F]

\[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {\left (f + g x\right )^{2}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}\, dx \]

[In]

integrate((g*x+f)**2/(a+b*ln(c*(e*x+d)**n))**3,x)

[Out]

Integral((f + g*x)**2/(a + b*log(c*(d + e*x)**n))**3, x)

Maxima [F]

\[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int { \frac {{\left (g x + f\right )}^{2}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((g*x+f)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="maxima")

[Out]

-1/2*((3*a*e^2*g^2 + (e^2*g^2*n + 3*e^2*g^2*log(c))*b)*x^3 + ((4*e^2*f*g + 5*d*e*g^2)*a + (2*e^2*f*g*n + d*e*g
^2*n + (4*e^2*f*g + 5*d*e*g^2)*log(c))*b)*x^2 + (d*e*f^2 + 2*d^2*f*g)*a + (d*e*f^2*n + (d*e*f^2 + 2*d^2*f*g)*l
og(c))*b + ((e^2*f^2 + 6*d*e*f*g + 2*d^2*g^2)*a + (e^2*f^2*n + 2*d*e*f*g*n + (e^2*f^2 + 6*d*e*f*g + 2*d^2*g^2)
*log(c))*b)*x + (3*b*e^2*g^2*x^3 + (4*e^2*f*g + 5*d*e*g^2)*b*x^2 + (e^2*f^2 + 6*d*e*f*g + 2*d^2*g^2)*b*x + (d*
e*f^2 + 2*d^2*f*g)*b)*log((e*x + d)^n))/(b^4*e^2*n^2*log((e*x + d)^n)^2 + b^4*e^2*n^2*log(c)^2 + 2*a*b^3*e^2*n
^2*log(c) + a^2*b^2*e^2*n^2 + 2*(b^4*e^2*n^2*log(c) + a*b^3*e^2*n^2)*log((e*x + d)^n)) + integrate(1/2*(9*e^2*
g^2*x^2 + e^2*f^2 + 6*d*e*f*g + 2*d^2*g^2 + 2*(4*e^2*f*g + 5*d*e*g^2)*x)/(b^3*e^2*n^2*log((e*x + d)^n) + b^3*e
^2*n^2*log(c) + a*b^2*e^2*n^2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8422 vs. \(2 (344) = 688\).

Time = 0.49 (sec) , antiderivative size = 8422, normalized size of antiderivative = 23.99 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((g*x+f)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="giac")

[Out]

1/2*b^2*e^2*f^2*n^2*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(e*x + d)^2/((b^5*e^3*n^5*log(e*x +
d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n
^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/n)) - b^2*d*e*f*g*n^2*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log
(e*x + d)^2/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) +
b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/n)) + 1/2*b^2*d^2*g^2*n^2*Ei(log(c)/n +
a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(e*x + d)^2/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*
log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/
n)) - 1/2*(e*x + d)*b^2*e^2*f^2*n^2*log(e*x + d)/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(
c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) - 2*(e*x
+ d)^2*b^2*e*f*g*n^2*log(e*x + d)/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^
3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) + (e*x + d)*b^2*d*e*f*g*
n^2*log(e*x + d)/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d
) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) - 3/2*(e*x + d)^3*b^2*g^2*n^2*log(e*x + d
)/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3
*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) + 2*(e*x + d)^2*b^2*d*g^2*n^2*log(e*x + d)/(b^5*e^3*n^5*
log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a
*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) - 1/2*(e*x + d)*b^2*d^2*g^2*n^2*log(e*x + d)/(b^5*e^3*n^5*log(e*x + d)^
2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*
log(c) + a^2*b^3*e^3*n^3) + 4*b^2*e*f*g*n^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(e*x + d))*e^(-2*a/(b*n))*log(e*x
+ d)^2/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*
e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(2/n)) - 4*b^2*d*g^2*n^2*Ei(2*log(c)/n + 2*a/(b
*n) + 2*log(e*x + d))*e^(-2*a/(b*n))*log(e*x + d)^2/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*
log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(2/
n)) + b^2*e^2*f^2*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(e*x + d)*log(c)/((b^5*e^3*n^5*log(e
*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*
e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/n)) - 2*b^2*d*e*f*g*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n)
)*log(e*x + d)*log(c)/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e
*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/n)) + b^2*d^2*g^2*n*Ei(log(c)/
n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(e*x + d)*log(c)/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(
e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n
^3)*c^(1/n)) - 1/2*(e*x + d)*b^2*e^2*f^2*n^2/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) +
2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) - (e*x + d)^2
*b^2*e*f*g*n^2/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d)
+ b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) + (e*x + d)*b^2*d*e*f*g*n^2/(b^5*e^3*n^5*lo
g(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b
^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) - 1/2*(e*x + d)^3*b^2*g^2*n^2/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4
*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*
e^3*n^3) + (e*x + d)^2*b^2*d*g^2*n^2/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4
*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) - 1/2*(e*x + d)*b^2*d
^2*g^2*n^2/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^
5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) + a*b*e^2*f^2*n*Ei(log(c)/n + a/(b*n) + log(e*x
+ d))*e^(-a/(b*n))*log(e*x + d)/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^
3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/n)) - 2*a*b*d*e*f*g
*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(e*x + d)/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^
4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3
*e^3*n^3)*c^(1/n)) + a*b*d^2*g^2*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(e*x + d)/((b^5*e^3*n
^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 +
2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/n)) + 9/2*b^2*g^2*n^2*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(e*x + d
))*e^(-3*a/(b*n))*log(e*x + d)^2/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^
3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(3/n)) - 1/2*(e*x + d)
*b^2*e^2*f^2*n*log(c)/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*
x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) - 2*(e*x + d)^2*b^2*e*f*g*n*log(c)/(
b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*lo
g(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) + (e*x + d)*b^2*d*e*f*g*n*log(c)/(b^5*e^3*n^5*log(e*x + d)^
2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*
log(c) + a^2*b^3*e^3*n^3) - 3/2*(e*x + d)^3*b^2*g^2*n*log(c)/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e
*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^
3) + 2*(e*x + d)^2*b^2*d*g^2*n*log(c)/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^
4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) - 1/2*(e*x + d)*b^2*
d^2*g^2*n*log(c)/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d
) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) + 8*b^2*e*f*g*n*Ei(2*log(c)/n + 2*a/(b*n)
+ 2*log(e*x + d))*e^(-2*a/(b*n))*log(e*x + d)*log(c)/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d
)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(
2/n)) - 8*b^2*d*g^2*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(e*x + d))*e^(-2*a/(b*n))*log(e*x + d)*log(c)/((b^5*e^3
*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2
+ 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(2/n)) + 1/2*b^2*e^2*f^2*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e
^(-a/(b*n))*log(c)^2/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*
x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/n)) - b^2*d*e*f*g*Ei(log(c)/n +
a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(c)^2/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c
) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/n)) +
1/2*b^2*d^2*g^2*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(c)^2/((b^5*e^3*n^5*log(e*x + d)^2 + 2*
b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c)
+ a^2*b^3*e^3*n^3)*c^(1/n)) - 1/2*(e*x + d)*a*b*e^2*f^2*n/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x
+ d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)
- 2*(e*x + d)^2*a*b*e*f*g*n/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4
*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) + (e*x + d)*a*b*d*e*f*g*n/(b^
5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(
c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) - 3/2*(e*x + d)^3*a*b*g^2*n/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b
^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c)
+ a^2*b^3*e^3*n^3) + 2*(e*x + d)^2*a*b*d*g^2*n/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c)
+ 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) - 1/2*(e*x
+ d)*a*b*d^2*g^2*n/(b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x +
d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3) + 8*a*b*e*f*g*n*Ei(2*log(c)/n + 2*a/(b*
n) + 2*log(e*x + d))*e^(-2*a/(b*n))*log(e*x + d)/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log
(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(2/n))
- 8*a*b*d*g^2*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(e*x + d))*e^(-2*a/(b*n))*log(e*x + d)/((b^5*e^3*n^5*log(e*x
+ d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^
3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(2/n)) + a*b*e^2*f^2*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(
c)/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n
^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/n)) - 2*a*b*d*e*f*g*Ei(log(c)/n + a/(b*n) + log(e
*x + d))*e^(-a/(b*n))*log(c)/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^
4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/n)) + a*b*d^2*g^2*Ei(lo
g(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(c)/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)
*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1
/n)) + 9*b^2*g^2*n*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(e*x + d))*e^(-3*a/(b*n))*log(e*x + d)*log(c)/((b^5*e^3*n^
5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2
*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(3/n)) + 4*b^2*e*f*g*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(e*x + d))*e^
(-2*a/(b*n))*log(c)^2/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e
*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(2/n)) - 4*b^2*d*g^2*Ei(2*log(c)/
n + 2*a/(b*n) + 2*log(e*x + d))*e^(-2*a/(b*n))*log(c)^2/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x +
d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c
^(2/n)) + 1/2*a^2*e^2*f^2*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))/((b^5*e^3*n^5*log(e*x + d)^2 + 2*
b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c)
+ a^2*b^3*e^3*n^3)*c^(1/n)) - a^2*d*e*f*g*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))/((b^5*e^3*n^5*lo
g(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b
^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/n)) + 1/2*a^2*d^2*g^2*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b
*n))/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3
*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(1/n)) + 9*a*b*g^2*n*Ei(3*log(c)/n + 3*a/(b*n) + 3
*log(e*x + d))*e^(-3*a/(b*n))*log(e*x + d)/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) +
2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(3/n)) + 8*a
*b*e*f*g*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(e*x + d))*e^(-2*a/(b*n))*log(c)/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^
5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) +
a^2*b^3*e^3*n^3)*c^(2/n)) - 8*a*b*d*g^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(e*x + d))*e^(-2*a/(b*n))*log(c)/((b
^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log
(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(2/n)) + 9/2*b^2*g^2*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(e*x
+ d))*e^(-3*a/(b*n))*log(c)^2/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*
n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(3/n)) + 4*a^2*e*f*g*Ei(
2*log(c)/n + 2*a/(b*n) + 2*log(e*x + d))*e^(-2*a/(b*n))/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x +
d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c
^(2/n)) - 4*a^2*d*g^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(e*x + d))*e^(-2*a/(b*n))/((b^5*e^3*n^5*log(e*x + d)^2
+ 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*lo
g(c) + a^2*b^3*e^3*n^3)*c^(2/n)) + 9*a*b*g^2*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(e*x + d))*e^(-3*a/(b*n))*log(c)
/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*log(e*x + d) + b^5*e^3*n^3
*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(3/n)) + 9/2*a^2*g^2*Ei(3*log(c)/n + 3*a/(b*n) + 3*log
(e*x + d))*e^(-3*a/(b*n))/((b^5*e^3*n^5*log(e*x + d)^2 + 2*b^5*e^3*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^3*n^4*l
og(e*x + d) + b^5*e^3*n^3*log(c)^2 + 2*a*b^4*e^3*n^3*log(c) + a^2*b^3*e^3*n^3)*c^(3/n))

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {{\left (f+g\,x\right )}^2}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3} \,d x \]

[In]

int((f + g*x)^2/(a + b*log(c*(d + e*x)^n))^3,x)

[Out]

int((f + g*x)^2/(a + b*log(c*(d + e*x)^n))^3, x)